Abstract
By using the familiar Stirling numbers, we derive the explicit forms of the weighted Newton-Cotes integration formulas and the weighted Adams-Bashforth and Adams-Moulton rules in terms of the Stirling numbers of the first and second kind. Some illustrative examples involving the Fredholm integral equation of the second kind are also considered.
Similar content being viewed by others
References
Atkinson, K.E.: A Survey of Numerical Methods for the Solution of Fredholm Integral Equations of the Second Kind. SIAM, Philadelphia (1976)
Butcher, J.C.: Numerical Methods for Ordinary Differential Equations. Wiley, New York (2008)
Butzer, P.L., Hauss, M.: On Stirling functions of the second kind. Stud. Appl. Math. 84, 71–91 (1991)
Gautschi, W.: Numerical Analysis: An Introduction. Birkhäuser, Boston (1997)
Kalogiratou, Z., Simos, T.E.: Newton-Cotes formulae for long-time integration. J. Comput. Appl. Math. 158, 75–82 (2003)
Masjed-Jamei, M., Milovanović, G.V., Jafari, M.A.: Explicit forms of weighted quardature rules with geometric nodes. Math. Comput. Model. 53, 1133–1139 (2011)
Masjed-Jamei, M., Milovanović, G.V., Jafari, M.A.: Closed expressions for coefficients in weighted Newton-Cotes quardatures. Filomat 27(4), 649–658 (2013)
Mastroianni, G., Milovanivić, G.V.: Interpolation Processes: Basic Theory and Applications. Springer Monographs in Mathematics. Springer-Verlag, Berlin (2008)
Milovanović, G.V., Cvetković, A.S.: Nonstandard Gaussian quadrature formulae based on operator values. Adv. Comput. Math. 32, 431–486 (2010)
Milovanović, G.V., Cvetković, A.S.: Gaussian quadrature rules using function derivatives. IMA J. Numer. Anal. 31, 358–377 (2011)
Srivastava, H.M.: Some families of generating functions associated with the Stirling numbers of the second kind. J. Math. Anal. Appl. 251, 752–769 (2000)
Srivastava, H.M., Choi, J.: Zeta and \(q\)-Zeta Functions and Associated Series and Integrals. Elsevier Science Publishers, Amsterdam (2012)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Masjed-Jamei, M., Jafari, M.A. & Srivastava, H.M. Some applications of the stirling numbers of the first and second kind. J. Appl. Math. Comput. 47, 153–174 (2015). https://doi.org/10.1007/s12190-014-0767-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-014-0767-4
Keywords
- Stirling numbers
- Explicit forms of the weighted Newton-Cotes integration formulas
- Explicit forms of the Adams-Bashforth and Adams-Moulton rules
- Newton interpolation
- Fredholm integral equations